Question

Let $A$ and $B$ be two $3\times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

• A. The system of linear equations $AX=0$ has a unique solution
• B. The system of linear equations $AX=0$ has infinitely many solutions
• C. $B$ is an invertible matrix
• D. adj (A) is an invertible matrix

Answer 1

Answer: (B)

$AB=0\Rightarrow |AB|=0$

If $|A|\ne 0,B=0$ (not possible)
If $|B|\ne 0,A=0$ (not possible)
Hence $|A|=|B|=0$
$\Rightarrow AX=0$ has infinitely many solutions